If $(F_n)$ is increasing and $\lim_{n\to\infty}\frac{F_1\dotsb F_n}{F_{n+1}}=0$ then $\sum\limits_{n=1}^\infty\frac1{F_n}$ is irrational

Question

Let $F_n$ be integers, and $F_1<F_2<\cdots<F_n<\cdots$. Suppose that $$\lim_{n\to\infty}\frac{F_1F_2\cdots F_{n-1}}{F_n}=0.$$ Prove then $$\sum_{n=1}^\infty \frac{1}{F_n}$$ is convergence, and the sum of which is irrational.

Answer 1

There are many exceptionally slow sequences, consider ${\prod_{i=1}^{n-1}F_i\over F_n}=\frac 1{\ln n}$ or even slower versions. But whatever sequence $1\over f(n)$ is chosen for the RHS, assuming $F_1$ is positive, we have $f(n)\to \infty$ as $n\to \infty$, thus

$$F_n=f(n)\prod_{i=1}^{n-1}F_i\ge \prod_{i=1}^{n-1}F_i$$

Then, assuming $\exists i:F_i\ge 2$, for all but a finite set of terms, we have

$$\sum_{i=k}^\infty\frac 1{F_i}\le \sum_{i=1}^\infty\frac 1{2^{2i}}$$

And thus $\sum_{i=1}^\infty \frac 1{F_i}$ converges under the specified assumptions.